Here \(m\)  is an integration constant (related to the mass) and \(a_N\)  has been defined in Eq. (REF ).
In this case (namely, for the solution with spherical symmetry in our family of solutions), the Einstein equations also lead to the constraint
\(p=q=1\)  in Eq. (REF ), therefore, this configuration does not have further parameters on top of the mass that can be considered as hair. This is different from the case of the flat black hole and the black string, as we will see in the next subsections.
Clearly, the above configuration represents a spherically symmetric black hole supported by pionic matter,
and it is the generalization to the \(SU(N)\)  case of the black hole found in Refs. [1]} (without the Skyrme term) and [2]}.
Also, the above solution is asymptotically the Anti-de Sitter version of the Barriola–Vilenkin space-time [3]}, and it generalizes the Schwarzschild
Anti-de Sitter space-time, which is recovered by setting \(K=0\) .
